Eigenvalue estimates for the basic Dirac operator on a Riemannian foliation admitting a basic harmonic 1-form

Abstract

On a compact Riemannian manifold M with a transverse spin foliation F of codimension q ≥ 3 , if M admits a non-trivial basic harmonic 1-form ω , then any eigenvalue λ of the basic Dirac operator satisfies the inequality λ 2 ≥ q − 1 4 ( q − 2 ) inf M ( σ ∇ + | κ | 2 ) , where σ ∇ is the transversal scalar curvature and κ is the mean curvature form of F . In the limiting case, F is minimal and ω is parallel.